Unremovable linked nodal structures protected by crystalline symmetries in stacked bilayer graphene with Kekulé texture

Linking structure is a new concept characterizing topological semimetals, which indicates the interweaving of gap-closing nodes at Fermi energy ($E_F$) with other below $E_F$. As number linked can be changed only via pair-creation or pair-annihilation, node more stable and robust than ordinary without linking. Here we propose type nodal between line (nodal surface) $E_F$ another in two-dimensional (three-dimensional) spinless fermion systems $\mathcal{IT}$ symmetry where $\mathcal{I}$ $\mathcal{T}$ indicate inversion time-reversal symmetries, respectively. Because additional chiral rotational our system, double band creates pair carrying same charges, thus are unremovable Lifshiftz transition, clearly distinct from cases reported previously. A realistic tight binding model effective theory developed for such linking structure. Also, using density functional calculations, class materials, composed stacked bilayer graphene Kekul\'{e} texture, as candidate system hosting